a red light winks every 6 seconds, a white one every 10 seconds, and a green one every 14 seconds. how often they wink together?
6 = 2*3
10 = 2*5
14 = 2*7
LCM(6,10,14) = 2*3*5*7 = 210
so, every 210 seconds they will blink at the same time
Well, it seems like those lights have quite the rhythm going on! To find out how often they wink together, we need to find the least common multiple of 6, 10, and 14.
So let's see, the prime factors for each number are:
6 = 2 * 3,
10 = 2 * 5,
14 = 2 * 7.
To find the least common multiple, we take the highest power of each prime factor that appears in any of the numbers. So we have:
2^1, 3^1, 5^1, and 7^1.
Multiplying them together, we get:
2 * 3 * 5 * 7 = 210.
Therefore, the red, white, and green lights will wink together every 210 seconds. That's a lot of winking!
To find out how often the lights wink together, we need to determine the time intervals at which all three lights coincide.
First, let's find the least common multiple (LCM) of 6, 10, and 14, which will give us the smallest number that is divisible by all three numbers.
The prime factors of 6 are 2 and 3.
The prime factors of 10 are 2 and 5.
The prime factors of 14 are 2 and 7.
We multiply the highest power of each prime factor, giving us:
2^1 * 3^1 * 5^1 * 7^1 = 2 * 3 * 5 * 7 = 210.
Therefore, the least common multiple (LCM) of 6, 10, and 14 is 210.
Thus, the three lights will wink together every 210 seconds.
To determine how often the red, white, and green lights wink together, we need to find the least common multiple (LCM) of the three intervals: 6 seconds, 10 seconds, and 14 seconds.
Here's how you can find the LCM:
1. List the multiples of each interval until you find a common multiple:
- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ...
- Multiples of 10: 10, 20, 30, 40, ...
- Multiples of 14: 14, 28, 42, ...
2. Look for the smallest common multiple from the lists:
- In this case, we can see that 42 seconds is the smallest common multiple.
Therefore, the red, white, and green lights will wink together every 42 seconds.